## An Investigation of Wave-Kinetic Theory: Hierarchy Equations, Phase Measure, and Resonance Singularity

##### Abstract

Wave-kinetic theory has been developed to describe the statistical dynamics of weakly nonlinear, dispersive waves. In the first part of this dissertation, we derive the wave-kinetic equations formally from a general model of Hamiltonian wave systems, in the standard limit of a continuum of weakly interacting dispersive waves with random phases. In this asymptotic limit we show that the correct dynamical equation for multi-mode amplitude distributions is not the well-known Peierls equation but is instead a reduced equation with only a subset of the terms in that equation. The equations that we derive are the direct analogue of the Boltzmann hierarchy obtained from the BBGKY hierarchy in the low-density limit for gases. We show that the asymptotic multi-mode equations possess factorized solutions for factorized initial data, which correspond to preservation in time of the property of “random phases & amplitudes”. The factors satisfy the equations for the 1-mode probability density functions (PDF’s) previously derived by Jakobsen & Newell and Choi et al. Analogous to the Klimontovich density in the kinetic theory of gases, we introduce the concepts of the “empirical spectrum” and the “empirical 1-mode PDF”. We show that the factorization of the hierarchy equations implies that these quantities are self-averaging: they satisfy the wave-kinetic equations of the spectrum and 1-mode PDF for almost any selection of phases and amplitudes from the initial ensemble. We show that both of these equations satisfy an H-theorem for an entropy defined by Boltzmann’s prescription S = k B log W. We also characterize the general solutions of our multi-mode distribution equations, for initial conditions with random phases but with no statistical assumptions on the amplitudes. Analogous to a result of Spohn for the Boltzmann hierarchy, these are “super-statistical solutions” that correspond to ensembles of solutions of the wave-kinetic equations with random initial conditions or random forces. On the basis of our results, we discuss possible kinetic explanations of intermittency and non-Gaussian statistics in wave turbulence. In particular, we advance the explanation of a “super-turbulence” produced by stochastic or turbulent solutions of the wave-kinetic equations themselves.
In the second part of the dissertation, we investigate a key assumption of wave-kinetic theory – dispersivity. We show that systems which are generally dispersive can have resonant sets of wave modes with identical group velocities, leading to a local breakdown of dispersivity. This shows up as a geometric singularity of the resonant manifold and possibly as an infinite phase measure in the collision integral. Such singularities occur widely for classical wave systems, including acoustical waves, Rossby waves, helical waves in rotating fluids, light waves in nonlinear optics and also in quantum transport, e.g. kinetics of electron-hole excitations (matter waves) in graphene. These singularities are the exact analogue of the critical points found by Van Hove in 1953 for phonon dispersion relations in crystals. The importance of these singularities in wave kinetics depends on the dimension of phase space D = (N − 2)d (d physical space dimension, N the number of waves in resonance) and the degree of degeneracy δ of the critical points. Following Van Hove, we show that non-degenerate singularities lead to finite phase measures for D > 2 but produce divergences when D ≤ 2 and possible breakdown of wave kinetics if the collision integral itself becomes too large (or even infinite). Similar divergences and possible breakdown can occur for degenerate singularities, when D − δ ≤ 2, as we find for several physical examples, including electron-hole kinetics in graphene. When the standard kinetic equation breaks down, one must develop a new singular wave kinetics. We discuss approaches from pioneering 1971 work of Newell & Aucoin on multi-scale perturbation theory for acoustic waves and field-theoretic methods based on exact Schwinger-Dyson integral equations for the wave dynamics.